A board audit will show what skills, experience and representation are going to be lacking; draw up a specification from this.
Circulate the specification as widely as possible, particularly targeting relevant websites and networks. State the process for selection and interview dates. For applications, some organisations request completion of a form but the more general practice at present seems to be a request for submission of a CV and covering letter. It's also useful to ask for referees.
You can set up a nominations committee of board members to oversee the process and conduct interviews, referring to the full board for approval of their choice(s). The CEO/director/artistic director should be involved in the process. Some organisations invite a prospective board member to attend a board meeting as observer prior to appointment.
Once a new member is selected, s/he should meet other board members and staff. It is helpful to be given an appointment letter, setting out duties and expectations.